My IQ has been increasing year by year.

Chapter 284 String Theory

Chen Zhuo sat at a round table by the window, on which sat a glass of water that had long since gone cold and several sheets of paper half-filled with writing.

The paper shows a discrete grid, with a continuous curved surface on the left and scattered points on the right.

The middle is empty.

Chen Zhuo looked at the blank space and sighed softly.

He put down his pen, leaned back in his chair, and rubbed his sore neck.

He had already decided on the general direction of reverting to discrete grid points yesterday in the corridor.

But after a night, when he actually tried to build the model on paper, he found himself stuck.

Continuous manifold, discrete mesh.

There is a huge gap between the two.

It's easy to break the continuous structure, but how can we prove that the broken mesh can still converge back to the original topology perfectly when taking the limit?

The bridge is broken.

Chen Zhuo sat quietly in the chair for a while. He glanced at the clock on the wall; it was 7:30 in the morning.

The Poincaré Conjecture presentation at the Harvard Science Center will begin at 9:00 AM.

He wasn't planning to go today.

He had been watching the meat grinder of partial differential equations in continuous space for three days, and he already understood the logic behind Professor Zhu and the others' deductions on stage.

Continuing to look at it will not help him overcome the bottleneck he is currently stuck in.

Chen Zhuo gathered the few sheets of paper on the table, folded them neatly, and put them in his bag. He stood up, went to the bathroom to wash his face, took his room key, and went out.

He planned to spend the next two days in a hotel or a quiet coffee shop trying to sort out the mess in his mind.

Once my thinking is completely clear, I'll go back to Princeton.

The hotel's buffet restaurant is on the second floor.

At this time, the restaurant was quite crowded, mostly with scholars from various fields wearing conference badges. They were carrying plates and moving around the food pick-up area, and the air was filled with the aroma of fried bacon, toasted bread, and coffee.

Chen Zhuo took a plate.

He picked up two slices of toast, took a small piece of butter, and scooped out a spoonful of scrambled eggs. As he passed the drinks section, he got a glass of warm milk.

Holding the plate, Chen Zhuo scanned the hall.

Pierre was sitting alone in a four-seater in the corner.

In front of Pierre was a cup of black tea and half a sandwich.

He wasn't eating; he was holding a few pages of a newly printed paper, looking at it with a frown, and there was a pen on the table next to him.

Chen Zhuo walked over and sat down in the seat opposite Pierre.

The plate touched the table with a soft sound.

Pierre looked up from the paper.

Good morning, teacher.

Chen Zhuo pushed the milk cup further in and greeted him.

"morning."

Pierre glanced at him and casually turned a page of the paper in his hand.

"Not with Li and the others?"

"No, Professor Li and Professor Zhu went to the classroom first."

Chen Zhuo used a knife to pick up a little butter and spread it evenly on the toast.

Pierre nodded, picked up the black tea beside him, and took a sip.

"Aren't you going?"

Pierre put down his cup and looked at Chen Zhuo.

"I'm not going."

Chen Zhuo took a bite of toast, chewed slowly, and swallowed.

The problems in continuous space are too complex; I'm not good at partial differential equations, and if I keep looking at them, my mind will get confused.

Pierre remained noncommittal. He picked up his pen and drew a horizontal line along the edge of the paper in his hand.

What's the date today?

Pierre suddenly asked a question.

Chen Zhuo thought for a moment.

"Number Five, what's wrong?"

Pierre put down his pen, folded up the few pages of the paper, and stuffed them into the bag next to him. He leaned back in his chair and looked at Chen Zhuo across from him.

Chen Zhu ate quietly, making very little noise as his knife and fork touched the plate.

His complexion looked good, without the withered look of someone who had stayed up for several nights in a row, but his brows were slightly furrowed.

Pierre was all too familiar with this situation.

"Is it stuck?"

Pierre spoke, his voice flat.

Chen Zhuo paused for a moment while cutting the eggs.

He looked up, met Pierre's gaze, and a slightly helpless smile appeared on his lips.

"Is it obvious?" Chen Zhuo put down his knife and fork.

"You made three cuts with your knife on the same egg, but in the end you still didn't put it in your mouth."

Pierre looked at the food on his plate.

"Did you get a shock from watching Zhu and Cao write on the blackboard in the lecture hall these past three days?"

Chen Zhuo shook his head.

"It's not exciting, it's just that the road is completely predictable."

Chen Zhuo grabbed a napkin from the side and picked up the pen that Pierre had placed on the table.

He removed the pen cap and drew an irregular circle on the left side of the napkin, and a few dots on the right side.

"What Professor Zhu and his team are doing is patching up this continuous circle."

Chen Zhuo tapped the circle with the tip of his pen.

"Forcibly stitching together the broken parts of the topology requires an astonishing amount of computation."

He moved the pen tip to a few dots on the right.

"I was thinking yesterday that if I want to bypass this huge amount of computation and grasp the characteristics of algebraic cycles, I have to go back and break the continuous manifold into discrete lattice points."

Pierre's gaze fell on the napkin, his eyes narrowing slightly.

As a top algebraic geometer, he understood the idea almost the instant Chen Zhuo drew those points.

"A very clever intuition," Pierre commented.

"But it won't work."

Chen Zhuo's voice was very soft. He held a pen and drew a dotted line between the circle on the left and the dot on the right. Then, he drew a heavy cross in the middle of the dotted line.

"I spent the whole night pushing it in my room last night."

Chen Zhuo looked at the cross on the paper.

"I've built the model, but I can't get past it."

Pierre remained silent, waiting for him to continue.

"There is a gap between continuous and discrete."

Chen Zhuo raised his eyelids and looked at Pierre.

"I've broken down the manifold, but when I take the limit at the end, how do I prove that these discrete meshes converge perfectly back to their original topological invariants?"

Chen Zhuo put down his pen and leaned back in his chair.

"The bridge is broken."

Chen Zhuo spread his hands, his tone revealing a genuine sense of defeat.

"As long as I cannot mathematically prove this convergence process rigorously, the discrete model I built is just a bunch of meaningless number games."

The restaurant remained noisy.

At the next table, several European scholars were loudly debating the eigenvalues ​​of a certain matrix, while a waiter pushed a food cart through the aisle and collected the empty plates.

The table in the corner was unusually quiet.

Pierre looked at the fork on the napkin.

He stared at it for a long time.

Chen Zhuo picked up his cup and finished the rest of the milk without urging him. He knew that Pierre was following his train of thought and deducing the next steps.

After a while, Pierre picked up the sandwich on the table and took a bite.

He chewed slowly, and after swallowing, he wiped his hands with a napkin.

You're getting stuck on one thing.

Pierre looked at Chen Zhuo and spoke.

Chen Zhuo was slightly taken aback.

"Mathematicians always have a damn fastidiousness when dealing with singularities."

Pierre's voice remained calm, but his words carried great weight.

"We always want to pursue absolute rigor from the very first step. All conditions must be met and all limits must be provable. Once a link in the chain breaks, we feel that the whole path is blocked."

Pierre tapped his finger on the napkin.

"This kind of fastidiousness is a virtue when it comes to simple things, but when faced with the most difficult problems of the century, it will trap you in the same place."

Chen Zhuo looked at Pierre's tapping fingers, lost in thought.

"You want to build a bridge that fits perfectly, but you don't even have a single brick right now."

Pierre shook his head.

"Do you think the Hodge conjecture will just stand there, waiting for you to polish your tools to perfection before you solve it?"

"Then what should we do?" Chen Zhuo asked.

Pierre did not answer the question.

He reached into his pocket, fumbled around, and pulled out a rectangular card.

He pushed the card in front of Chen Zhuo.

Chen Zhuo lowered his head.

It was a pass with the Harvard University logo printed on it, against a background of a starry sky and several complex physics formulas.

"Harvard University Physics Department, Jefferson Physics Laboratory"

Chen Zhuo read the words on the card aloud.

He looked up at Pierre with some confusion.

"At 2 p.m., there will be an internal workshop on string theory at Jefferson Lab."

Pierre picked up the glass of cold red tea.

"I was originally invited to sit in on the meeting, but I have a meeting this afternoon and can't get away. You go instead."

Chen Zhuo picked up the pass and looked at it in his hand.

"String theory?" Chen Zhuo smiled.

"Teacher, I don't even want to look at partial differential equations. You want me to look at how physicists work with eleven-dimensional spacetime?"

"Didn't you say you didn't know how to build a bridge?"

Pierre put down his coffee cup.

"You should go and see how those physicists do this dirty work."

Chen Zhuo's smile faded somewhat.

Pierre leaned back in his chair.

"When theoretical physicists encounter intractable continuous spacetime singularities in their calculations, do you think they'll stop and try to prove that damn limit exists, like mathematicians do?"

Pierre gave a cold laugh.

"They won't. They'll just close their eyes and brutally shred spacetime into a grid, using the most violent discrete nodes to forcibly replace continuous spatial evolution."

Chen Zhuo's heart skipped a beat.

He looked at Pierre.

"Physicists don't care about absolute mathematical rigor; as long as the calculation results conform to physical laws and experimental observations, they dare to use them."

Pierre looked into Chen Zhuo's eyes.

"Their methods are riddled with flaws within a purely mathematical system, and are outrageously crude."

Pierre paused for a moment, then leaned forward slightly.

"But it worked."

Chen Zhuo didn't speak; he looked down at the laboratory pass in his hand.

It simply and crudely shreds space and time.

There's no need to worry about whether the limits are rigorous.

These words were like a hammer, striking his tangled thoughts directly.

Just like when he was under the tutelage of the sorcerer.

I think I've been getting a bit stuck in a rut lately.

Pierre, observing his reaction, knew he understood.

"It's not that you don't have blueprints, you just think they're not clean enough."

Pierre picked up the pen on the table and drew several horizontal lines at the break in the dotted line on the napkin.

"There's a ready-made scaffold at the physics department. It's covered in mud, but it can stand on the cliff."

Pierre threw down his pen.

"Go and see how they laid out the mesh, extract their skeleton, and remove those loose physical shells."

Pierre stood up and straightened the hem of his suit jacket.

"What's left is the bridge you need. As for how to build it securely using algebraic geometry, that's something you'll have to worry about when you get back."

Chen Zhuo looked up at Pierre, the sense of defeat in his eyes fading somewhat.

He put the pass into his coat pocket.

"I see."

Chen Zhuo nodded.

"Thank you for your help, teacher."

"This is just a suggestion; whether you can see any benefit from it is up to you."

Pierre picked up the bag.

"Take your time exploring Boston these next couple of days. There's no rush to go back to Princeton."

After saying that, Pierre didn't linger and turned to leave the restaurant.

Chen Zhuo sat in his seat.

He looked at the napkin with lines drawn on it on the table.

The big X on the paper doesn't look so rigid anymore; instead, there are several horizontal lines next to it.

Rough scaffolding.

Chen Zhuo took the toast he hadn't finished eating, put it in his mouth, and slowly ate it.

He picked up the glass of milk and drank it in one gulp.

Chen Zhuo stood up, pushed the chair back to its original position, gathered the scraps of paper on the table and the napkin with the drawing on it together, crumpled it into a ball, and threw it into the trash can he was passing by.

It's still early; there are still several hours until 2 PM.

He decided to go back to his room and first look up some basic information about lattice gauge field theory.

If you don't do your homework beforehand, you'll most likely not understand the string theory seminar you attend in the afternoon.